y=a(x-h)^2+kVariables
- (h,k): vertex
- a: vertical scale and opening direction
How to use this formula
Expresses a parabola using its vertex and vertical scale.
Important notes
- a>0 opens upward; a<0 opens downward.
Quick example
y=2(x−3)²+1 has vertex (3,1).
Applicability, worked calculation, and verification
Domain and applicability
Apply Quadratic Vertex Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- a>0 opens upward; a<0 opens downward.
- For the Quadratic Vertex Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Quadratic Vertex Form when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Quadratic Vertex Form, check zero, negative, and extreme input values before relying on the result.
- When using Quadratic Vertex Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form y=a(x-h)^2+k for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For y=2(x-3)²+1, the vertex is (3,1). At x=5, y=2(2²)+1=9.
- Read h=3 and k=1 from a(x-h)²+k.
- Use the positive coefficient a=2 to identify an upward-opening parabola.
- Substitute x=5 to obtain y=9.
Independent verification
Substituting x=3 gives the minimum y=1, confirming the stated vertex.
Common mistakes
- Before substituting values into Quadratic Vertex Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Quadratic Vertex Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Quadratic Vertex Form in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
y=a(x-h)^2+k. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Vertex Form.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Quadratic Vertex Form used for?
Expresses a parabola using its vertex and vertical scale.
Can I copy this formula as LaTeX?
Yes. Copy y=a(x-h)^2+k or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.